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• W2007457286 abstract "Given a redundant dictionary Φ, represented by an M×N matrix (Φ∈RM×N) and a target signal y∈RM, the sparse approximation problem asks to find an approximate representation of y using a linear combination of at most k atoms. This note presents a hardness result for sparse approximation problem under a measure of quality, which is essentially the squared multiple correlation in statistical analysis. We show that unless P=NP, all polynomial time algorithms which provide a k-sparse vector x should satisfy‖ΦxΦx+y‖22⩽(1−1e)‖Φx⁎Φx⁎+y‖22, where x⁎ is the optimal k-sparse solution and Φx denotes the column sub-matrix of Φ which consists of the column vectors with indices of non-zero elements in x. We relate this result to the recent algorithmic results of Das and Kempe (2008, 2011) [1], [2] and conclude that Forward Regression and Orthogonal Matching Pursuit are almost the best one can hope for solving the sparse approximation problem under the squared multiple correlation metric, especially when the dictionary is near orthogonal." @default.
• W2007457286 created "2016-06-24" @default.
• W2007457286 creator A5087909897 @default.
• W2007457286 date "2013-07-01" @default.
• W2007457286 modified "2023-09-23" @default.
• W2007457286 title "A note on the hardness of sparse approximation" @default.
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• W2007457286 doi "https://doi.org/10.1016/j.ipl.2013.04.014" @default.
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